not disconcert us. Most laws dealing with the macroscopic behavior of matter are really statistical laws whose validity follows from the random behavior of huge num-bers of molecules. For example, in thermodynamics, we refer to the pressure P of a system. The pressure a gas exerts on the container walls results from the collisions of molecules with the walls. There is a possibility that, at some instant, the gas molecules might all be moving inward toward the interior of the container, so that the gas would exert zero pressure on the container. Likewise, the molecular motion at a given instant might make the pressure on some walls differ significantly from that on other walls.
However, such situations are so overwhelmingly improbable that we can with com-plete confidence ascribe a single uniform pressure to the gas.
Chapter 3
The Second Law of Thermodynamics 104
gravitational attractions. There was a possibility that gravitational attractions might eventually overcome the expansion, thereby causing the universe to begin to contract, ultimately bringing all matter together again. Perhaps a new Big Bang would then ini-tiate a new cycle of expansion and contraction. An alternative possibility was that there was not enough matter to prevent the expansion from continuing forever.
If the cyclic expansion–contraction cosmological model is correct, what will hap-pen in the contraction phase of the universe? If the universe returns to a state essentially the same as the initial state that preceded the Big Bang, then the entropy of the universe would decrease during the contraction phase. This expectation is further supported by the arguments for a direct connection between the thermodynamic and cosmological arrows of time. But what would a universe with decreasing entropy be like? Would time run backward in a contracting universe? What is the meaning of the statement that
“time runs backward”?
Astronomical observations made in 1998 and subsequent years have shown the startling fact that the rate of expansion of the universe is increasing with time, rather than slowing down as formerly believed. The accelerated expansion is driven by a mysterious entity called dark energy, hypothesized to fill all of space. Observations indicate that ordinary matter constitutes only about 4% of the mass–energy of the universe. Another 22% is dark matter, whose nature is unknown (but might be as yet undiscovered uncharged elementary particles). The existence of dark matter is in-ferred from its observed gravitational effects. The remaining 74% of the universe is dark energy, whose nature is unknown. The ultimate fate of the universe depends on the nature of dark energy, and what is now known about it seems to indicate that the expansion will likely continue forever, but this is not certain. For discussion of the possibilities for the ultimate fate of the universe and how these possibilities depend on the properties of dark energy, see R. Vaas, “Dark Energy and Life’s Ultimate Future,” arxiv.org/abs/physics/0703183.
3.9 SUMMARY
We assumed the truth of the Kelvin–Planck statement of the second law of ther-modynamics, which asserts the impossibility of the complete conversion of heat to work in a cyclic process. From the second law, we proved that dqrev/T is the differ-ential of a state function, which we called the entropy S. The entropy change in a process from state 1 to state 2 is S 兰21dqrev/T, where the integral must be eval-uated using a reversible path from 1 to 2. Methods for calculating S were dis-cussed in Sec. 3.4.
We used the second law to prove that the entropy of an isolated system must increase in an irreversible process. It follows that thermodynamic equilibrium in an isolated system is reached when the system’s entropy is maximized. Since isolated systems spontaneously change to more probable states, increasing entropy corre-sponds to increasing probability p. We found that S k ln p a, where the Boltzmann constant k is k R/NAand a is a constant.
Important kinds of calculations dealt with in this chapter include:
• Calculation of S for a reversible process using dS dqrev/T.
• Calculation of S for an irreversible process by finding a reversible path between the initial and final states (Sec. 3.4, paragraphs 5, 7, and 9).
• Calculation of S for a reversible phase change using S H/T.
• Calculation of S for constant-pressure heating using dS dqrev/T (CP/T) dT.
• Calculation of S for a change of state of a perfect gas using Eq. (3.30).
• Calculation of S for mixing perfect gases at constant T and P using Eq. (3.33).
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FURTHER READING
Denbigh, pp. 21–42, 48–60; Kestin, chap. 9; Zemansky and Dittman, chaps. 6, 7, 8.
PROBLEMS
Problems
105
Section 3.2
3.1 True or false? (a) Increasing the temperature of the hot reservoir of a Carnot-cycle engine must increase the efficiency of the engine. (b) Decreasing the temperature of the cold reser-voir of a Carnot-cycle engine must increase the efficiency of the engine. (c) A Carnot cycle is by definition a reversible cycle.
(d) Since a Carnot cycle is a cyclic process, the work done in a Carnot cycle is zero.
3.2 Consider a heat engine that uses reservoirs at 800°C and 0°C. (a) Calculate the maximum possible efficiency. (b) If qHis 1000 J, find the maximum value of w and the minimum value of qC.
3.3 Suppose the coldest reservoir we have at hand is at 10°C.
If we want a heat engine that is at least 90% efficient, what is the minimum temperature of the required hot reservoir?
3.4 A Carnot-cycle heat engine does 2.50 kJ of work per cycle and has an efficiency of 45.0%. Find w, qH, and qCfor one cycle.
3.5 Heat pumps and refrigerators are heat engines running in reverse; a work input w causes the system to absorb heat qC from a cold reservoir at TCand emit heat qHinto a hot reser-voir at TH. The coefficient of performance K of a refrigerator is qC/w, and the coefficient of performance e of a heat pump is
qH/w. (a) For reversible Carnot-cycle refrigerators and heat pumps, express K and e in terms of TCand TH. (b) Show that erevis always greater than 1. (c) Suppose a reversible heat pump transfers heat from the outdoors at 0°C to a room at 20°C. For each joule of work input to the heat pump, how much heat will be deposited in the room? (d) What happens to Krevas TCgoes to 0 K?
3.6 Use sketches of the work wbydone by the system for each step of a Carnot cycle to show that wbyfor the cycle equals the area enclosed by the curve of the cycle on a P-V plot.
Section 3.4
3.7 True or false? (a) A change of state from state 1 to state 2 produces a greater increase in entropy when carried out irre-versibly than when done reirre-versibly. (b) The heat q for an ir-reversible change of state from state 1 to 2 might differ from the heat for the same change of state carried out reversibly. (c) The higher the absolute temperature of a system, the smaller the in-crease in its entropy produced by a given positive amount dqrev of reversible heat flow. (d) The entropy of 20 g of H2O(l) at 300 K and 1 bar is twice the entropy of 10 g of H2O(l) at 300 K and 1 bar. (e) The molar entropy of 20 g of H2O(l) at 300 K and
1 bar is equal to the molar entropy of 10 g of H2O(l) at 300 K and 1 bar. ( f ) For a reversible isothermal process in a closed system, S must be zero. (g) The integral 兰21 T1CV dT in Eq. (3.30) is always equal to CVln (T2/T1). (h) The system’s en-tropy change for an adiabatic process in a closed system must be zero. (i) Thermodynamics cannot calculate S for an irre-versible process. ( j) For a reirre-versible process in a closed system, dq is equal to T dS. (k) The formulas of Sec. 3.4 enable us to calculate S for various processes but do not enable us to find the value of S of a thermodynamic state.
3.8 The molar heat of vaporization of Ar at its normal boiling point 87.3 K is 1.56 kcal/mol. (a) Calculate S for the vapor-ization of 1.00 mol of Ar at 87.3 K and 1 atm. (b) Calculate S when 5.00 g of Ar gas condenses to liquid at 87.3 K and 1 atm.
3.9 Find S when 2.00 mol of O2is heated from 27°C to 127°C with P held fixed at 1.00 atm. Use CP,mfrom Prob. 2.48.
3.10 Find S for the conversion of 1.00 mol of ice at 0°C and 1.00 atm to 1.00 mol of water vapor at 100°C and 0.50 atm. Use data from Prob. 2.49.
3.11 Find S when 1.00 mol of water vapor initially at 200°C and 1.00 bar undergoes a reversible cyclic process for which q 145 J.
3.12 Calculate S for each of the following changes in state of 2.50 mol of a perfect monatomic gas with CV,m 1.5R for all temperatures: (a) (1.50 atm, 400 K) → (3.00 atm, 600 K);
(b) (2.50 atm, 20.0 L) → (2.00 atm, 30.0 L); (c) (28.5 L, 400 K)
→ (42.0 L, 400 K).
3.13 For N2(g), CP,mis nearly constant at 29.1 J/(mol K) for temperatures in the range 100 K to 400 K and low or moderate pressures. Find S for the reversible adiabatic compression of 1.12 g of N2(g) from 400 torr and 1000 cm3to a final volume of 250 cm3. Assume perfect-gas behavior.
3.14 Find S for the conversion of 10.0 g of supercooled water at 10°C and 1.00 atm to ice at 10°C and 1.00 atm.
Average cPvalues for ice and supercooled water in the range 0°C to 10°C are 0.50 and 1.01 cal/(g °C), respectively. See also Prob. 2.49.
3.15 State whether each of q, w, U, and S is negative, zero, or positive for each step of a Carnot cycle of a perfect gas.
3.16 After 200 g of gold [cP 0.0313 cal/(g °C)] at 120.0°C is dropped into 25.0 g of water at 10.0°C, the system is allowed to reach equilibrium in an adiabatic container. Find (a) the final temperature; (b) SAu; (c) (d) ¢SH ¢SAu ¢SH2O.
2O;
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106
3.17 Calculate S for the mixing of 10.0 g of He at 120°C and 1.50 bar with 10.0 g of O2at 120°C and 1.50 bar.
3.18 A system consists of 1.00 mg of ClF gas. A mass spec-trometer separates the gas into the species 35ClF and 37ClF.
Calculate S. Isotopic abundances: 19F 100%; 35Cl 75.8%;
37Cl 24.2%.
3.19 Let an isolated system be composed of one part at T1and a second part at T2, with T2 T1; let the parts be separated by a wall that allows heat flow at only an infinitesimal rate. Show that, when heat dq flows irreversibly from T2to T1, we have dS dq/T1 dq/T2(which is positive). (Hint: Use two heat reservoirs to carry out the change of state reversibly.)
Section 3.5
3.20 True or false? (a) For a closed system, S can never be negative. (b) For a reversible process in a closed system, S must be zero. (c) For a reversible process in a closed system,
Sunivmust be zero. (d) For an adiabatic process in a closed system, S cannot be negative. (e) For a process in an isolated system, S cannot be negative. ( f ) For an adiabatic process in a closed system, S must be zero. (g) An adiabatic process can-not decrease the entropy of a closed system. (h) For a closed sys-tem, equilibrium has been reached when S has been maximized.
3.21 For each of the following processes deduce whether each of the quantities S and Suniv is positive, zero, or negative.
(a) Reversible melting of solid benzene at 1 atm and the normal melting point. (b) Reversible melting of ice at 1 atm and 0°C.
(c) Reversible adiabatic expansion of a perfect gas. (d) Revers-ible isothermal expansion of a perfect gas. (e) Adiabatic expan-sion of a perfect gas into a vacuum (Joule experiment).
( f ) Joule–Thomson adiabatic throttling of a perfect gas.
(g) Reversible heating of a perfect gas at constant P. (h) Revers-ible cooling of a perfect gas at constant V. (i) Combustion of benzene in a sealed container with rigid, adiabatic walls. ( j) Adi-abatic expansion of a nonideal gas into vacuum.
3.22 (a) What is S for each step of a Carnot cycle? (b) What is Sunivfor each step of a Carnot cycle?
3.23 Prove the equivalence of the Kelvin–Planck statement and the entropy statement [the set-off statement after Eq. (3.40)]
of the second law. [Hint: Since the entropy statement was de-rived from the Kelvin–Planck statement, all we need do to show the equivalence is to assume the truth of the entropy statement and derive the Kelvin–Planck statement (or the Clausius state-ment, which is equivalent to the Kelvin–Planck statement) from the entropy statement.]
Section 3.6
3.24 Willard Rumpson (in later life Baron Melvin, K.C.B.) defined a temperature scale with the function f in (3.43) as
“take the square root” and with the water triple-point tempera-ture defined as 200.00°M. (a) What is the temperatempera-ture of the steam point on the Melvin scale? (b) What is the temperature of the ice point on the Melvin scale?
3.25 Let the Carnot-cycle reversible heat engine A absorb heat q3per cycle from a reservoir at t3and discard heat q2A
per cycle to a reservoir at t2. Let Carnot engine B absorb heat q2Bper cycle from the reservoir at t2and discard heat q1per cycle to a reservoir at t1. Further, let q2A q2B, so that engine B absorbs an amount of heat from the t2reservoir equal to the heat deposited in this reservoir by engine A. Show that
where the function g is defined as 1 erev. The heat reservoir at t2can be omitted, and the combination of engines A and B can be viewed as a single Carnot engine operating between t3 and t1; hence g(t1, t3) q1/q3. Therefore
(3.59)
Since t3does not appear on the left side of (3.59), it must can-cel out of the numerator and denominator on the right side.
After t3is canceled, the numerator takes the form f(t1) and the denominator takes the form f(t2), where f is some function;
we then have
(3.60)
which is the desired result, Eq. (3.42). [A more rigorous de-rivation of (3.60) from (3.59) is given in Denbigh, p. 30.]
3.26 For the gaussian probability distribution, the probability of observing a value that deviates from the mean value by at least x standard deviations is given by the following infinite series (M. L. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Natl. Bur. Stand. Appl. Math. Ser. 55, 1964, pp. 931–932):
where the series is useful for reasonably large values of x.
(a) Show that 99.7% of observations lie within 3 standard de-viations from the mean. (b) Calculate the probability of a devi-ation 106standard deviations.
3.27 If the probability of observing a certain event in a single trial is p, then clearly the probability of not observing it in one trial is 1 p. The probability of not observing it in n indepen-dent trials is then (1 p)n; the probability of observing it at least once in n independent trials is 1 (1 p)n. (a) Use these ideas to verify the calculation of Eq. (3.58). (b) How many times must a coin be tossed to reach a 99% probability of observing at least one head?
General
3.28 For each of the following sets of quantities, all the quan-tities except one have something in common. State what they have in common and state which quantity does not belong with the others. (In some cases, more than one answer for the prop-erty in common might be possible.) (a) H, U, q, S, T; (b) T, S, q, w, H; (c) q, w, U, U, V, H; (d) r, Sm, M, V; (e) H, S, dV, P; ( f ) U, V, H, S, T.
2
22p ex2>2a1 x 1
x3 3 x5 # # #b g1t1, t22 f1t12
f1t22 g1t1, t22 g1t1, t32
g1t2, t32 g1t2, t32g1t1, t22 q1>q3
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3.29 Estimate the volume of cooling water used per minute by a 1000-MW power plant whose efficiency is 40%. Assume that the cooling water undergoes a 10°C temperature rise (a typical value) when it cools the steam.
3.30 A certain perfect gas has CV,m a bT, where a 25.0 J/(mol K) and b 0.0300 J/(mol K2). Let 4.00 mol of this gas go from 300 K and 2.00 atm to 500 K and 3.00 atm. Calculate each of the following quantities for this change of state. If it is impossible to calculate a quantity from the given information, state this. (a) q; (b) w; (c) U; (d) H; (e) S.
3.31 Classify each of these processes as reversible or irre-versible: (a) freezing of water at 0°C and 1 atm; (b) freezing of supercooled water at 10°C and 1 atm; (c) burning of carbon in O2to give CO2at 800 K and 1 atm; (d ) rolling a ball on a floor with friction; (e) the Joule–Thomson experiment; ( f ) adi-abatic expansion of a gas into vacuum (the Joule experiment);
(g) use of a frictionless piston to infinitely slowly increase the pressure on an equilibrium mixture of N2, H2, and NH3, thereby shifting the equilibrium.
3.32 For each of the following pairs of systems, state which system (if either) has the greater U and which has the greater S.
(a) 5 g of Fe at 20°C and 1 atm or 10 g of Fe at 20°C and 1 atm;
(b) 2 g of liquid water at 25°C and 1 atm or 2 g of water vapor at 25°C and 20 torr; (c) 2 g of benzene at 25°C and 1 bar or 2 g of benzene at 40°C and 1 bar; (d ) a system consisting of 2 g of metal M at 300 K and 1 bar and 2 g of M at 310 K and 1 bar or a system consisting of 4 g of M at 305 K and 1 bar. Assume the specific heat of M is constant over the 300 to 310 K range and the volume change of M is negligible over this range; (e) 1 mol of a perfect gas at 0°C and 1 atm or 1 mol of the same perfect gas at 0°C and 5 atm.
3.33 Which of these cyclic integrals must vanish for a closed system with P-V work only? (a) 养 P dV; (b) 养 (P dV V dP);
(c)养 V dV; (d) 养 dqrev/T; (e) 养 H dT; ( f ) 养 dU; (g) 养 dqrev; (h)养 dqP; (i) 养 dwrev; ( j) 养 dwrev/P.
3.34 Consider the following quantities: CP, CP,m, R (the gas constant), k (Boltzmann’s constant), q, U/T. (a) Which have the same dimensions as S? (b) Which have the same dimen-sions as Sm?
3.35 What is the relevance to thermodynamics of the following refrain from the Gilbert and Sullivan operetta H.M.S. Pinafore?
“What, never? No, never! What, never? Well, hardly ever!”
107 3.36 In the tropics, water at the surface of the ocean is warmer than water well below the surface. Someone proposes to draw heat from the warm surface water, convert part of it to work, and discard the remainder to cooler water below the surface.
Does this proposal violate the second law?
3.37 Use (3.15) to show that it is impossible to attain the ab-solute zero of temperature.
3.38 Suppose that an infinitesimal crystal of ice is added to 10.0 g of supercooled liquid water at 10.0°C in an adiabatic container and the system reaches equilibrium at a fixed pressure of 1 atm. (a) What is H for the process? (b) The equilibrium state will contain some ice and will therefore consist either of ice plus liquid at 0°C or of ice at or below 0°C. Use the answer to (a) to deduce exactly what is present at equilibrium. (c) Calcu-late S for the process. (See Prob. 2.49 for data.)
3.39 Give the SI units of (a) S; (b) Sm; (c) q; (d ) P; (e) Mr (molecular weight); ( f ) M (molar mass).
3.40 Which of the following statements can be proved from the second law of thermodynamics? (a) For any closed system, equilibrium corresponds to the position of maximum entropy of the system. (b) The entropy of an isolated system must remain constant. (c) For a system enclosed in impermeable adiabatic walls, the system’s entropy is maximized at equilibrium. (d ) The entropy of a closed system can never decrease. (e) The entropy of an isolated system can never decrease.
3.41 True or false? (a) For every process in an isolated sys-tem, T 0. (b) For every process in an isolated system that has no macroscopic kinetic or potential energy, U 0. (c) For every process in an isolated system, S 0. (d) If a closed sys-tem undergoes a reversible process for which V 0, then the P-V work done on the system in this process must be zero.
(e)S when 1 mol of N2(g) goes irreversibly from 25°C and 10 L to 25°C and 20 L must be the same as S when 1 mol of N2(g) goes reversibly from 25°C and 10 L to 25°C and 20 L.
( f ) S 0 for every adiabatic process in a closed system.
(g) For every reversible process in a closed system, S H/T.
(h) A closed-system process that has T 0, must have U 0. (i) For every isothermal process in a closed system, S
H/T. (j) q 0 for every isothermal process in a closed system.
(k) In every cyclic process, the final and initial states of the sys-tem are the same and the final and initial states of the sur-roundings are the same.
R3.1 For a closed system, give an example of each of the fol-lowing. If it is impossible to have an example of the process, state this. (a) An isothermal process with q 0. (b) An adia-batic process with T 0. (c) An isothermal process with
U 0. (d) A cyclic process with S 0. (e) An adiabatic process with S 0. ( f ) A cyclic process with w 0.
R3.2 State what experimental data you would need to look up to calculate each of the following quantities. Include only the minimum amount of data needed. Do not do the calculations.
(a) U and H for the freezing of 653 g of liquid water at 0°C and 1 atm. (b) S for the melting of 75 g of Na at 1 atm and its normal melting point. (c) U and H when 2.00 mol of O2gas